Band Structure of (D|E) Cells

Assume the two materials in Fig. 3.1 are both homogeneous along y-direction (i.e.

) and the probe signal wave travels in the (...D|E|D|E…) structure always along x-direction. In the section that follows, the reflectance on the left side interface ( )of such an EIT-based periodic medium, which is in fact a 1D -layer (D|E) layered structure bounded by the GaAs dielectric material, will be addressed. In this thesis, the lattice

/ y 0

  

0 x N

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constants of the (D|E) cells are chosen asa b 0.1μm.

In order to show how sensitive (to the probe frequency) the band structure of the EIT photonic crystal is, let us first see the dispersive relation of the 1D infinite periodic (D|E) cells, in which the probe frequency _p is far from the resonant frequency of the atomic level |1 | 3 transition. Here we choose the typical atomic transition frequency

15 31 5.0 10

   s^1, and the thickness of the two layers a0.1 10 ^6m (GaAs dielectric) and b0.1 10 ^6 m(EIT medium). As the probe frequency detuning of TE waves in Fig.

3.2 is quite large ( _p c/with   a b), the strong dispersion of EIT cannot be exhibited, and the present (D|E) layered structure behaves like a conventional 1D photonic crystal. However, when the probe frequency detuning _papproaches zero (or negligibly small compared with c/having the order of magnitude10 s^{15 1}, e.g., _p is tuned onto resonance, i.e.,   _{p c}that equals 1.0 10 s ^{7 1}), it would exhibit a band with fine structure (and hence remarkable frequency-sensitive reflectance and transmittance). The band structure in the probe frequency detuning range    _p/ ₃ [ 2.5 10 , 2.5 10 ] ⁸   ⁸ is plotted in Fig. 3.2(a). The typical atomic and optical parameters such as the atomic number density N_a , the electrical dipole moment₃₁ , the control frequency detuning _c, the spontaneous emission decay rate ₃ and the dephasing rate ₂ are chosen exactly the same as in Figs. 2.2 and Fig. 2.3 (i.e.   ₃ 2 10 s^{7 1} and ₂ 0.005   ₃ 1 10 s^{5 1} and these typical parameters are also tabled in table 2 on page 24). The Rabi frequency of the control field is  _c 2.0 10 ⁷ s^1. Since, seen from Fig. 3.2(a), there are some fine structures of the band in the frequency range (   _p/ ₃ [ 2.5 10 , 2.5 10 ] ⁸   ⁸ ) that need to be addressed, we present the intricate structures in Fig. 3.2(b)-(d) and demonstrate them in more details. In Fig. 3.2(b), for example, as the probe frequency detuning tends to the resonant frequency (  _{p c}, i.e.,  _p/ ₃ approaches almost zero compared withc/), both the real and imaginary parts of the Bloch wave number K would arise, because the strong dispersion of EIT medium, of which the relative refractive index is a complex number, plays a key role for creating such a band structure.

In Fig. 3.2(c) we have shown the fine structure of the band induced by the EIT resonance.

However, the detailed fine texture cannot be signified by the course curves in Fig. 3.2(c),

21

since the band structure is plotted within the large range of probe frequency detuning.

(a) (b)

(c) (d)

Fig. 3.2 The bandgap structure of the 1D infinite periodic (D|E) cells when the probe frequency of TE waves is far from the resonance. In (a) is the band structure in the probe frequency detuning range   _p/ ₃  2.5 10 , 2.5 10 ⁸   ⁸. In (b), (c) and (d) are the fine detailsexhibited in the EIT-based band structure in the probe frequency detuning ranges (in units of₃), i.e.,   _p/ ₃  1.5 10 , 1.5 10 ⁷   ⁷

7 7

12 10 , 2.0 10

    

 ^{, and} 2.5 10 , 0.0 10⁸   ⁸,respectively.

We shall in what follows treat further the fine structure of the band of EIT-based photonic crystal when the |1 | 3 transition of the EIT atomic levels is on resonance. It follows from Fig. 3.3(a) that after aligning dielectric GaAs side by side with EIT medium there are three extreme values of imaginary part K in the Bloch wave number (or Kk n_{0 r,(D|E)} ).

Coincidently, there are also three extreme values of real partKin the Bloch wave number (or

0 r,(D|E)

Kk n ). However, neither of them reaches the band edge K0 or K0.5 (in the

22

units of 2 / ). Note thatKand Ksimultaneously exist, since the refractive index of the EIT medium has an imaginary part when the probe frequency is tuned onto resonance of |1 | 3 transition of the EIT atomic system. It should be emphasized that the band structure (i.e., K

and Kvary as the probe frequency detuning _pchanges slightly) is very sensitive to the probe frequency detuning. From Fig. 3.3 (a) one can see that both the real and imaginary parts of the Bloch wave number change drastically between 0 and 0.5 (in the units of 2 / ) in a very narrow frequency band (namely, a very small change, e.g., at the level of one part in 10⁸ in the probe frequency, gives rise to a large variation in the Bloch wave number). In particular, the slope (dK d/ _p) is almost divergent at the position  _p/ ₃ 0.5. The reason for this is because

 _p 0.5 ₃ is exactly the two-photon resonant frequency (  _{p c}). As there is almost divergent dispersion close to  _p 0.5 ₃, the effects of slow light and the negative group velocity in such an EIT-based periodic layered material deserve consideration. This would lead to promising applications in designing devices for slowing down light speed. Besides, the EIT-based band structure is tunable in response to the intensity (characterized by the control Rabi frequency _c) of the external control field, since the refractive index of the EIT medium can be controlled by the control field. In Fig. 3.3 (b) the real part of the Bloch wave number K decreasesas_c increases from 0 to4 ₃, and then increases when   _c/ ₃ 4 , and the imaginary part of the Bloch wave number increases first in the range   _c/ ₃ [0,1] and then decreases when   _c/ ₃ 1. This, therefore, means that we can use one optical field to manipulate the wave propagation of the other optical field via such an EIT-based periodic layered structure.

As we have shown the characteristics of both sensitivity and tunability of the EIT-based band structure in Fig. 3.2(b) and Fig. 3.3, we shall present its three-dimensional behavior as both the probe frequency detuning and the Rabi frequency of control field vary. The sensitivity and the tunability versus the probe frequency detuning _p and the Rabi frequency _c of control field, respectively, are shown in Fig. 3.4.

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(a) (b)

Fig. 3.3 The Bloch wave number K of the 1D infinite periodic (D|E) cells when the EIT atomic transition is on resonance. The curves in (a) indicate the real and imaginary parts of the normalized Bloch wave number K sensitive to the probe frequency detuning _p , where the Rabi frequency of the control field is chosen as

7 1

c 2 10 s^

   . The curves in (b) show the tunable Bloch wave number K at the frequency detuning   _{p 2 10 s}^{7 1} when the Rabi frequency _c of the control field changes.

(a) (b)

Fig. 3.4 The real part (a) and the imaginary part (b) of the normalized Bloch wave number K (in the units of 2 / ) of the 1D infinite periodic (D|E) cells versus

pand _c.Both the real and imaginary parts oftheBloch wave number K are sensitive to the small change in the probe frequency (the slopedK d/ _p of the dispersive curve is much more larger than that in a conventional photonic

24

crystal), andboth thereal and imaginary parts of the Bloch wave number at any fixed probe frequencies can becontrollable by the control Rabi frequency_c.

Table 2 Typical physical parameters setting for peridical (D|E) layered structure

Parameter Value Unit Parameter Value Unit

n 1 1/3.54 - n 2 nEIT n

 

a 0.1 m b 0.1 m

3 _{2 10} ⁷ _s^1 ₂ 1 10 ⁵ s^1

p _{2 10} ⁷/10⁸ s^1 c





N a _{5 10} ²⁰ m^3 31 _{1 10} ^29 C m

N 1 ~ 6 - ph _{0 s}^1

31 _{5 10} ¹⁵ _s^1 _{p  5 10}¹⁵ _s^1

32 _{ 5 10}¹⁵ _s^1 _{c  5 10}¹⁵ _s^1

c 32 c

  









In the curve of the reflectance Rin the narrow band close to  _p 0.5 ₃ increases as the total layer number N increases, that is to say, the more layers there are in the (D|E) cell structure, the more drastic change there is in the reflectance and transmittance on the left interface of this periodic layered structure.

25

(a) (b)

(c) (d)

(e) (f)

Fig. 3.5 The reflection coefficient of the 1D N -layer periodic (D|E) cells as _pchanges.

(a) N 1, (b) N 2, (c) N 3, (d) N 4, (e) N 5, (f) N 6.

26

(a) (b)

(c) (d)

(e) (f)

Fig. 3.6 The reflectance and transmittance on the left interface of the 1D N-layer periodic (D|E) cells as_pchanges. (a)N 1, (b) N2, (c) N3, (d) N4, (e) N 5,(f) N6. The reflectance drops to its minimum value and the transmittance reaches itsmaximum value close to the resonant frequency

 _p 0.5 ₃. As the total layer numberN of the (D|E) cells increases, the extreme value increases also, but all of the extreme value allocations are all

27

packed within a gradually narrower band of  _p 0.3 ₃~ 0.5 ₃ .The extreme value allocation number is expected to be proportional to the number

N of layers.

The fine structure of the reflectance for showing extraordinary sensitivity to the frequency of the probe field is demonstrated in Fig. 3.7. Here we plot only the reflectance of two cases (N 3 andN 6) as an illustrative example. It can be seen that some oscillations in the curve are exhibited in the resonant frequency rangeΔ_p[0.35,0.55] ₃. Apparently, such a frequency-sensitive behavior in the light wave transmission can have some interesting applications. We expect that the mechanism of sensitive optical response in such an EIT-based periodic layered medium would be employed in fabrication of many new quantum optical and photonic devices such as photonic logic gates, optical switches, photonic transistor as well as wavelength sensor. Clearly, the device of wavelength sensor designed based on such an EIT-based periodic layered medium can, in principle, find a practical application in the field of color matching and sorting, where, for example, precise measurements of wavelengths or frequencies of optical fields are required.

(a) (b)

Fig. 3.7 The fine details of two typical cases for showing sensitivity of the reflectance to the frequency of the probe field in a narrow bandwidth. (a)N3, (b) N 6. The reflectance on the left interface of the periodic (D|E) cells oscillates as the probe frequency detuning Δ_pchanges slightly (at the level of one part in10⁸ in the probe frequency_p).

We have shown that the EIT band structure and its reflectance and transmittance are quite

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sensitive to the probe frequency, since the microscopic electric polarizability as well as the electric permittivity of the EIT medium are caused by the atomic energy level transition processes from the ground state to the excited states, in which the quantum interference relevant to the atomic phase coherence is involved. The tunable and sensitive optical response induced by quantum interference of the multilevel atomic system has several useful applications, such as in the design of many new photonic devices. The present scheme can be generalized to a four-level EIT system, where two control fields and one probe field drive the atomic level transitions. Obviously, the band structure in such an EIT photonic crystal would be more sensitive to the probe frequency than in a three-level EIT photonic crystal presented in this paper. Apart from this intriguing property, there are more interesting applications that we can suggest based on the four-level EIT photonic crystal, e.g., some examples of photonic devices, in which the two control fields and the probe field act as the input and output signals, respectively, can be designed. These include the devices of functional and logic operations, such as AND, OR, NOT, NAND, NOR, EXOR and EXNOR gates. Besides, the strong dispersion in the susceptibility found here can lead to dramatic modification in the speed of light that propagates through the EIT-based periodic layered medium. We expect that all these new optical properties relevant to quantum coherence, including their applications to photonic devices, could be realized experimentally in the near future.

A periodic layered medium with unit cells composed of dielectric (e.g., GaAs) and EIT (electromagnetically induced transparency) atomic vapor is suggested and the frequency-sensitive optical response via tunable band structure in this EIT-based layered medium is considered. As the quantum interference that is induced by external control field makes an essential contribution to the optical response such a periodic layered medium shows a flexible behavior that is sensitive to frequency. The controllable band structure that depends on the external control field can be applicable to designs of new devices such as photonic switches and photonic logic gates, where one laser field can be controlled by the other one, and would have potential applications in the field of integrated optical circuits and other related areas, e.g., the all-optical technique.

Over the past three decades, artificial electromagnetic materials that can exhibit peculiar optical properties have attracted considerable attention in various scientific and technological areas. A number of recent theoretical and experimental works have demonstrated that the effects of phase coherence control in multilevel atomic systems and

29

semiconductor quantum dots can lead to novel quantum optical phenomena of nearly resonant light, including atomic coherent population trapping (CPT) , laser without inversion (LWI), electromagnetically induced transparency (EIT) and some new applications relevant to EIT. EIT that is one of the most important quantum coherent effects arises from quantum interference in energy level transitions from the ground states to the excited ones. It is such a quantum optical phenomenon that if only one resonant laser beam propagates through a medium, it will be absorbed; but if there are two resonant laser beams propagating in the same medium, neither would be absorbed, and hence the opaque medium would become a transparent one. Apart from the nearly zero absorption at probe resonant frequency, quantum coherent effect in an EIT medium (e.g., atomic vapor or semiconductor-quantum-dot material) can also give rise to a strong dispersion near resonance and a tunable manipulation of light propagation (e.g., as the optical properties of the EIT medium depend on the external control field intensity, it can be used to realize beam focusing and hence EIT lensing). Since it can exhibit many intriguing optical properties and effects, EIT could be applied to a variety of areas of optics, and enable us to achieve novel results, e.g., some unusual physical effects associated with EIT include the ultraslow light pulse propagation, the superluminal light propagation, the light storage in atomic vapors, some of which are expected to be beneficial (and powerful) for developing new technologies in quantum optics and photonics.